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Mirrors > Home > MPE Home > Th. List > eldprdi | Structured version Visualization version GIF version |
Description: The domain of definition of the internal direct product, which states that 𝑆 is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.) |
Ref | Expression |
---|---|
eldprdi.0 | ⊢ 0 = (0g‘𝐺) |
eldprdi.w | ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } |
eldprdi.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
eldprdi.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
eldprdi.3 | ⊢ (𝜑 → 𝐹 ∈ 𝑊) |
Ref | Expression |
---|---|
eldprdi | ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldprdi.1 | . 2 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
2 | eldprdi.3 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊) | |
3 | eqid 2825 | . . 3 ⊢ (𝐺 Σg 𝐹) = (𝐺 Σg 𝐹) | |
4 | oveq2 6918 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝐺 Σg 𝑓) = (𝐺 Σg 𝐹)) | |
5 | 4 | rspceeqv 3544 | . . 3 ⊢ ((𝐹 ∈ 𝑊 ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝐹)) → ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)) |
6 | 2, 3, 5 | sylancl 580 | . 2 ⊢ (𝜑 → ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)) |
7 | eldprdi.2 | . . 3 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
8 | eldprdi.0 | . . . 4 ⊢ 0 = (0g‘𝐺) | |
9 | eldprdi.w | . . . 4 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } | |
10 | 8, 9 | eldprd 18764 | . . 3 ⊢ (dom 𝑆 = 𝐼 → ((𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)))) |
11 | 7, 10 | syl 17 | . 2 ⊢ (𝜑 → ((𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)))) |
12 | 1, 6, 11 | mpbir2and 704 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∃wrex 3118 {crab 3121 class class class wbr 4875 dom cdm 5346 ‘cfv 6127 (class class class)co 6910 Xcixp 8181 finSupp cfsupp 8550 0gc0g 16460 Σg cgsu 16461 DProd cdprd 18753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-1st 7433 df-2nd 7434 df-ixp 8182 df-dprd 18755 |
This theorem is referenced by: dprdfsub 18781 dprdf11 18783 dprdsubg 18784 dprdub 18785 dpjidcl 18818 |
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